Bernoulli polynomials

In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, occur in the study of many special functions and, in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.

A similar set of polynomials, based on a similar generating function, is the family of Euler polynomials.

Representations

The Bernoulli polynomials Bn admit a variety of different representations. Which among them should be taken to be the definition may depend on one's purposes.

Explicit formula

$B_{n}(x)=\sum _{k=0}^{n}{n \choose k}B_{n-k}x^{k},$ $E_{m}(x)=\sum _{k=0}^{m}{m \choose k}{\frac {E_{k}}{2^{k}}}\left(x-{\frac {1}{2}}\right)^{m-k}\,.$ for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers. These formulas can be obtained from the translation formulas below.

Generating functions

The generating function for the Bernoulli polynomials is

${\frac {te^{xt}}{e^{t}-1}}=\sum _{n=0}^{\infty }B_{n}(x){\frac {t^{n}}{n!}}.$ The generating function for the Euler polynomials is

${\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x){\frac {t^{n}}{n!}}.$ Representation by a differential operator

The Bernoulli polynomials are also given by

$B_{n}(x)={D \over e^{D}-1}x^{n}$ where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that

$\int _{a}^{x}B_{n}(u)~du={\frac {B_{n+1}(x)-B_{n+1}(a)}{n+1}}~.$ cf. integrals below. By the same token, the Euler polynomials are given by

$E_{n}(x)={\frac {2}{e^{D}+1}}x^{n}.$ Representation by an integral operator

The Bernoulli polynomials are the unique polynomials determined by

$\int _{x}^{x+1}B_{n}(u)\,du=x^{n}.$ $(Tf)(x)=\int _{x}^{x+1}f(u)\,du$ on polynomials f, simply amounts to

{\begin{aligned}(Tf)(x)={e^{D}-1 \over D}f(x)&{}=\sum _{n=0}^{\infty }{D^{n} \over (n+1)!}f(x)\\&{}=f(x)+{f'(x) \over 2}+{f''(x) \over 6}+{f'''(x) \over 24}+\cdots ~.\end{aligned}} This can be used to produce the inversion formulae below.

Another explicit formula

An explicit formula for the Bernoulli polynomials is given by

$B_{m}(x)=\sum _{n=0}^{m}{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x+k)^{m}.$ Note the remarkable similarity to the globally convergent series expression for the Hurwitz zeta function. Indeed, one has

$B_{n}(x)=-n\zeta (1-n,x)$ where ζ(s, q) is the Hurwitz zeta; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n.

The inner sum may be understood to be the nth forward difference of xm; that is,

$\Delta ^{n}x^{m}=\sum _{k=0}^{n}(-1)^{n-k}{n \choose k}(x+k)^{m}$ where Δ is the forward difference operator. Thus, one may write

$B_{m}(x)=\sum _{n=0}^{m}{\frac {(-1)^{n}}{n+1}}\,\Delta ^{n}x^{m}.$ This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals

$\Delta =e^{D}-1$ where D is differentiation with respect to x, we have, from the Mercator series

${D \over e^{D}-1}={\log(\Delta +1) \over \Delta }=\sum _{n=0}^{\infty }{(-\Delta )^{n} \over n+1}.$ As long as this operates on an mth-degree polynomial such as xm, one may let n go from 0 only up to m.

An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, which follows from the expression as a finite difference.

An explicit formula for the Euler polynomials is given by

$E_{m}(x)=\sum _{n=0}^{m}{\frac {1}{2^{n}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x+k)^{m}\,.$ The above follows analogously, using the fact that

${\frac {2}{e^{D}+1}}={\frac {1}{1+\Delta /2}}=\sum _{n=0}^{\infty }{\Bigl (}-{\frac {\Delta }{2}}{\Bigr )}^{n}.$ Sums of pth powers

Using either the above integral representation of $x^{n}$ or the identity $B_{n}(x+1)-B_{n}(x)=nx^{n-1}$ , we have

$\sum _{k=0}^{x}k^{p}=\int _{0}^{x+1}B_{p}(t)\,dt={\frac {B_{p+1}(x+1)-B_{p+1}}{p+1}}$ (assuming 00 = 1). See Faulhaber's formula for more on this.

The Bernoulli and Euler numbers

The Bernoulli numbers are given by $\textstyle B_{n}=B_{n}(0).$ This definition gives $\textstyle \zeta (-n)={\frac {(-1)^{n}}{n+1}}B_{n+1}$ for $\textstyle n=0,1,2,\ldots$ .

An alternate convention defines the Bernoulli numbers as $\textstyle B_{n}=B_{n}(1).$ The two conventions differ only for $n=1$ since $B_{1}(1)={\tfrac {1}{2}}=-B_{1}(0)$ .

The Euler numbers are given by $E_{n}=2^{n}E_{n}({\tfrac {1}{2}}).$ Explicit expressions for low degrees

The first few Bernoulli polynomials are:

{\begin{aligned}B_{0}(x)&=1\\[8pt]B_{1}(x)&=x-{\frac {1}{2}}\\[8pt]B_{2}(x)&=x^{2}-x+{\frac {1}{6}}\\[8pt]B_{3}(x)&=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{2}}x\\[8pt]B_{4}(x)&=x^{4}-2x^{3}+x^{2}-{\frac {1}{30}}\\[8pt]B_{5}(x)&=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{3}}x^{3}-{\frac {1}{6}}x\\[8pt]B_{6}(x)&=x^{6}-3x^{5}+{\frac {5}{2}}x^{4}-{\frac {1}{2}}x^{2}+{\frac {1}{42}}.\end{aligned}} The first few Euler polynomials are:

{\begin{aligned}E_{0}(x)&=1\\[8pt]E_{1}(x)&=x-{\frac {1}{2}}\\[8pt]E_{2}(x)&=x^{2}-x\\[8pt]E_{3}(x)&=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{4}}\\[8pt]E_{4}(x)&=x^{4}-2x^{3}+x\\[8pt]E_{5}(x)&=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{2}}x^{2}-{\frac {1}{2}}\\[8pt]E_{6}(x)&=x^{6}-3x^{5}+5x^{3}-3x.\end{aligned}} Maximum and minimum

At higher n, the amount of variation in Bn(x) between x = 0 and x = 1 gets large. For instance,

$B_{16}(x)=x^{16}-8x^{15}+20x^{14}-{\frac {182}{3}}x^{12}+{\frac {572}{3}}x^{10}-429x^{8}+{\frac {1820}{3}}x^{6}-{\frac {1382}{3}}x^{4}+140x^{2}-{\frac {3617}{510}}$ which shows that the value at x = 0 (and at x = 1) is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. D.H. Lehmer showed that the maximum value of Bn(x) between 0 and 1 obeys

$M_{n}<{\frac {2n!}{(2\pi )^{n}}}$ unless n is 2 modulo 4, in which case

$M_{n}={\frac {2\zeta (n)n!}{(2\pi )^{n}}}$ (where $\zeta (x)$ is the Riemann zeta function), while the minimum obeys

$m_{n}>{\frac {-2n!}{(2\pi )^{n}}}$ unless n is 0 modulo 4, in which case

$m_{n}={\frac {-2\zeta (n)n!}{(2\pi )^{n}}}.$ These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.

Differences and derivatives

The Bernoulli and Euler polynomials obey many relations from umbral calculus:

$\Delta B_{n}(x)=B_{n}(x+1)-B_{n}(x)=nx^{n-1},$ $\Delta E_{n}(x)=E_{n}(x+1)-E_{n}(x)=2(x^{n}-E_{n}(x)).$ (Δ is the forward difference operator). Also,

$E_{n}(x+1)+E_{n}(x)=2x^{n}.$ These polynomial sequences are Appell sequences:

$B_{n}'(x)=nB_{n-1}(x),$ $E_{n}'(x)=nE_{n-1}(x).$ Translations

$B_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}B_{k}(x)y^{n-k}$ $E_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}E_{k}(x)y^{n-k}$ These identities are also equivalent to saying that these polynomial sequences are Appell sequences. (Hermite polynomials are another example.)

Symmetries

$B_{n}(1-x)=(-1)^{n}B_{n}(x),\quad n\geq 0,$ $E_{n}(1-x)=(-1)^{n}E_{n}(x)$ $(-1)^{n}B_{n}(-x)=B_{n}(x)+nx^{n-1}$ $(-1)^{n}E_{n}(-x)=-E_{n}(x)+2x^{n}$ $B_{n}\left({\frac {1}{2}}\right)=\left({\frac {1}{2^{n-1}}}-1\right)B_{n},\quad n\geq 0{\text{ from the multiplication theorems below.}}$ Zhi-Wei Sun and Hao Pan  established the following surprising symmetry relation: If r + s + t = n and x + y + z = 1, then

$r[s,t;x,y]_{n}+s[t,r;y,z]_{n}+t[r,s;z,x]_{n}=0,$ where

$[s,t;x,y]_{n}=\sum _{k=0}^{n}(-1)^{k}{s \choose k}{t \choose {n-k}}B_{n-k}(x)B_{k}(y).$ Fourier series

The Fourier series of the Bernoulli polynomials is also a Dirichlet series, given by the expansion

$B_{n}(x)=-{\frac {n!}{(2\pi i)^{n}}}\sum _{k\not =0}{\frac {e^{2\pi ikx}}{k^{n}}}=-2n!\sum _{k=1}^{\infty }{\frac {\cos \left(2k\pi x-{\frac {n\pi }{2}}\right)}{(2k\pi )^{n}}}.$ Note the simple large n limit to suitably scaled trigonometric functions.

This is a special case of the analogous form for the Hurwitz zeta function

$B_{n}(x)=-\Gamma (n+1)\sum _{k=1}^{\infty }{\frac {\exp(2\pi ikx)+e^{i\pi n}\exp(2\pi ik(1-x))}{(2\pi ik)^{n}}}.$ This expansion is valid only for 0  x  1 when n  2 and is valid for 0 < x < 1 when n = 1.

The Fourier series of the Euler polynomials may also be calculated. Defining the functions

$C_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\cos((2k+1)\pi x)}{(2k+1)^{\nu }}}$ and

$S_{\nu }(x)=\sum _{k=0}^{\infty }{\frac {\sin((2k+1)\pi x)}{(2k+1)^{\nu }}}$ for $\nu >1$ , the Euler polynomial has the Fourier series

$C_{2n}(x)={\frac {(-1)^{n}}{4(2n-1)!}}\pi ^{2n}E_{2n-1}(x)$ and

$S_{2n+1}(x)={\frac {(-1)^{n}}{4(2n)!}}\pi ^{2n+1}E_{2n}(x).$ Note that the $C_{\nu }$ and $S_{\nu }$ are odd and even, respectively:

$C_{\nu }(x)=-C_{\nu }(1-x)$ and

$S_{\nu }(x)=S_{\nu }(1-x).$ They are related to the Legendre chi function $\chi _{\nu }$ as

$C_{\nu }(x)=\operatorname {Re} \chi _{\nu }(e^{ix})$ and

$S_{\nu }(x)=\operatorname {Im} \chi _{\nu }(e^{ix}).$ Inversion

The Bernoulli and Euler polynomials may be inverted to express the monomial in terms of the polynomials.

Specifically, evidently from the above section on integral operators, it follows that

$x^{n}={\frac {1}{n+1}}\sum _{k=0}^{n}{n+1 \choose k}B_{k}(x)$ and

$x^{n}=E_{n}(x)+{\frac {1}{2}}\sum _{k=0}^{n-1}{n \choose k}E_{k}(x).$ Relation to falling factorial

The Bernoulli polynomials may be expanded in terms of the falling factorial $(x)_{k}$ as

$B_{n+1}(x)=B_{n+1}+\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}(x)_{k+1}$ where $B_{n}=B_{n}(0)$ and

$\left\{{\begin{matrix}n\\k\end{matrix}}\right\}=S(n,k)$ denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:

$(x)_{n+1}=\sum _{k=0}^{n}{\frac {n+1}{k+1}}\left[{\begin{matrix}n\\k\end{matrix}}\right]\left(B_{k+1}(x)-B_{k+1}\right)$ where

$\left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)$ denotes the Stirling number of the first kind.

Multiplication theorems

The multiplication theorems were given by Joseph Ludwig Raabe in 1851:

For a natural number m1,

$B_{n}(mx)=m^{n-1}\sum _{k=0}^{m-1}B_{n}\left(x+{\frac {k}{m}}\right)$ $E_{n}(mx)=m^{n}\sum _{k=0}^{m-1}(-1)^{k}E_{n}\left(x+{\frac {k}{m}}\right)\quad {\mbox{ for }}m=1,3,\dots$ $E_{n}(mx)={\frac {-2}{n+1}}m^{n}\sum _{k=0}^{m-1}(-1)^{k}B_{n+1}\left(x+{\frac {k}{m}}\right)\quad {\mbox{ for }}m=2,4,\dots$ Integrals

Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:

• $\int _{0}^{1}B_{n}(t)B_{m}(t)\,dt=(-1)^{n-1}{\frac {m!n!}{(m+n)!}}B_{n+m}\quad {\text{for }}m,n\geq 1$ • $\int _{0}^{1}E_{n}(t)E_{m}(t)\,dt=(-1)^{n}4(2^{m+n+2}-1){\frac {m!n!}{(m+n+2)!}}B_{n+m+2}$ Periodic Bernoulli polynomials

A periodic Bernoulli polynomial Pn(x) is a Bernoulli polynomial evaluated at the fractional part of the argument x. These functions are used to provide the remainder term in the Euler–Maclaurin formula relating sums to integrals. The first polynomial is a sawtooth function.

Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions.

The following properties are of interest, valid for all $x$ :

{\begin{aligned}&P_{k}(x){\text{ is continuous for all }}k\neq 1\\[5pt]&P_{k}'(x){\text{ exists and is continuous for }}k=0,k\geq 3\\[5pt]&P'_{k}(x)=kP_{k-1}(x),k\geq 3\end{aligned}} 